FOM: Connections between mathematics, physics and FOM
Jeffrey John Ketland
Jeffrey.Ketland at nottingham.ac.uk
Mon Jan 31 11:20:08 EST 2000
Connections between maths, physics and fom: continued
Mark Steiner and Matt Install have raised some questions about
my earlier posting. Joe Shipman has sent me an excellent paper of
his about relations between physics and computability theory. So I
have composed this point-by-point discussion of some of the topics
raised. In particular, I want to say that there is fom significance in
many of the topics below. There are also open research problems
(most of which I lack the detailed requisite skill to work on).
(A) Plato’s dictum: Mathematics is a science of abstract patterns
and structures
(B) Gardner’s dictum: The physical universe is mathematically
structured
(C) Galileo’s dictum: the book of Nature is written in the language
of mathematics
(D) Field’s program: Eliminating all (!) mathematics from science
(E) Feferman’s program: Eliminating non-constructive mathematics
from science
(F) The Kazhdan problem: the difference between interesting and
uninteresting
mathematical structures.
(G) The Wigner problem: The “unreasonable applicability” problem
(H) The Kronecker problem: The finiteness of the human mind.
(I) The Laplace-Poincare problem: predictability and computational
tractability.
(J) Quantum computation
(K) The Zeno Problem: supertasks in spacetime and infinity
machines
(L) Godelian phenomena in physics: Reverse physics?
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(A) Plato’s dictum: Mathematics is a science of abstract patterns
and structures
This is the slogan of the contemporary structuralists in the
philosophy of mathematics, Michael Resnik and Stewart Shapiro. I
am in broad agreement with this (realist) position. See:
[1] Michael Resnik 1997: Mathematics as a Science of Patterns
[2] Stewart Shapiro 1997: Philosophy of Mathematics – Structure
and Ontology
At Matt Insall correctly observed, the English word “pattern” has a
connotation of predictability, computational tractability or whatever.
That is not what is usually meant here. A pattern can be any
pattern at all – a random binary sequence, for example. And a
structure can be any abstract structure at all, including non-
axiomatizable structures.
The crucial point is that these abstract structures exist outside us,
are not “made” or “constructed” by us. They have an objective
existence independent of the human mind, and even independently
of their exemplification in the physical universe. So Plato’s dictum
is meant to be realist or platonist slogan, to be chanted against
constructivists and formalists.
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(B) Gardner’s dictum: The physical universe is mathematically
structured
This is meant to provide a rough explanation for the very possibility
of applying mathematics (never mind applying which bit of maths to
which bit of the world). Again, the idea goes way back to Plato and
before (Pythagoras, of course).
Mathematical structures are Plato’s “Forms” and the physical
systems are the “imperfect copies” of those Forms.
Not all Forms need to have instances in the world. Church’s
famous example, the concept or form “purple cow” has no
instances in the world. Similarly, not all mathematical structures
need have instances in the physical world. For a start, some
mathematical structures are just too big (like the natural
models of ZFC)!
I think I disagree with Mark Steiner that Gardner’s dictum is a
pseudo-explanation. Really, it’s just a very weak statement saying
that some of the abstract structures that pure mathematicians
study really do turn up in physics. It certainly doesn’t explain,
for example, why spacetime is a four dimensional manifold or why
the gluon wavefunctions are irreducible representations of SU(3)!
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(C) Galileo’s dictum: the book of Nature is written in the language
of mathematics
Although (A) and (B) don’t solve all of the problems to do with
applicability, they do explain Galileo’s dictum. Physical particles
move along continuous curves in continuous spacetime. Their
velocities are derivatives of their position functions and their
accelerations are proportional to the local forces acting. So, the
book of Nature will contain lots of analysis and reference to real
numbers, and functions thereon.
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(D) Field’s program: Eliminating all (!) mathematics from science
The Quine-Putnam argument is a major hot topic in contemporary
philosophy of mathematics. It says that our scientific theories
make endless reference to mathematical entities, so we can’t be
realists about those theories without also being realist about those
mathematical entities. Quine says such things in tens of his
articles and books, starting with
[3] W.V. Quine 1948: “On What There is”, in Quine 1980, From a
Logical Point of View.
and finishing with
[4] W.V. Quine 1995: From Stimulus to Science
For Putnam’s version of the QP argument, see
[5] Hilary Putnam 1971: “Philosophy of Logic”, in Putnam 1979,
Mathematics, Matter and Method: Collected Papers.
In 1980 Hartry Field transformed philosophy of mathematics by
attacking the QP argument at its heart and claiming that all (!!!) of
mathematics can be eliminated from science. See:
[6] Hartry Field 1980: Science Without Numbers
Field’s 2 main claims are that
(i) we can always eliminate mathematics from any physical
theory to obtain a “nominalistic” theory (i.e., no quantifiers
ranging over numbers, or sets).
(ii) if we add mathematics to a nominalistic theory, we always
get a conservative extension.
He substantiated these two claims by, indeed, giving a nominalistic
version of Newtonian gravitational physics (including a nominalistic
or synthetic treatment of spacetime) and by arguing that any model
of a nominalistic theory N can be expanded to a model of the result
of adding ZFC to N (if this were true, then N+ZFC would have to be
a conservative extension of N).
Field’s point about conservative extensions is closely connected to
Hilbert’s ideas about “real” and “ideal” mathematics. PRA is
supposed to codify the “real” and adding “ideal” set theory should
give a conservative extension. Then “ideal” set theory would just be
a “useful, but dispensable, instrument” for finding out real things
about the numbers. Similarly, Field’s idea is that adding
mathematics (i.e., set theory) would just be “useful, but
dispensable, instrument” for finding things out about the concrete
world.
HOWEVER. Damn!!! It doesn’t work! Adding set theory to PRA is
non-conservative and adding set theory to certain synthetic
descriptions of spacetime is non-conservative. This is closely
connected to Godel’s theorems. In the mathematical case, all this
is well-known to readers of this list. But the case for the non-
conservativeness of adding set theory to nominalistic spacetime
theories is carefully developed in:
[7] Stewart Shapiro 1983 “Conservativeness and Incompleteness”,
Journal of Philosophy 80.
And in
[8] John Burgess and Gideon Rosen 1997: A Subject With No
Object.
In fact, the Introduction to Burgess and Rosen’s book above [8] is
the best introductory discussion of modern nominalism that I know
of.
Matt Insall makes a point which Stewart Shapiro also made (in the
book [2] above) about Field’s program. Even if one could eliminate
direct reference to numbers and sets from physical theory, we
would still be studying a *structure exemplified in the world*.
Aside from the conservativeness issue, there are many technical
problems in this program. How do you do the nominalizing for
Quantum Theory? For General Relativity? My view is that Field’s
program is beautiful and philosophically attractive.
But I think it doesn’t work.
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(E) Feferman’s program: Eliminating non-constructive mathematics
from science
Solomon Feferman wrote a paper in 1988
[9] Solomon Feferman 1988: “Weyl Vindicated: Das Continuum 70
years later” (Temi e propetive della logica e della filosopfia della
scienza contemporanee, Bologna).
developing a predicative set theory W (conservative over PA),
finishing with an argument that all the mathematics needed for
science could be developed within W.
Not many philosophers know much about this idea: that although
we’ll use mathematics in science, we’re restricted to some kind of
constructive mathematics. In fact, I’m not sure that many people
have worked on it at all.
However, there are some interesting papers by Geoffrey Hellman
arguing that certain more restrictive versions of constructivism are
demonstrably inadequate to the needs of present science. In
particular,
[10] Geoffrey Hellman 1993: “Gleason’s theorem is not
constructively provable”, J.Philosophical Logic.
[11] Geoffrey Hellman 1998: “Mathematical constructivism in
spacetime”, British Journal for the Philosophy of Science.
There was a storm of protest by a few constructivists about the
claim about Gleason’s theorem’s unprovability. I don’t know the
details. The points he makes in [11] are more convincing, however.
According to GR, the (spacetime) continuum is right “out
there”, exemplified in physical spacetime. So, we are permitted to
use non-constructive analysis to study this genuinely real physical
system. An example Hellman discusses is the proof of Hawking-
Penrose singularity theorems, which are non-constructive
existence theorems.
As well as Feferman’s program (based on predicative foundations),
there would also be “Bishop’s program”, and perhaps many others,
corresponding to the various foundational positions. My own view is
that just as constructive mathematics is inadequate for the needs
of core mathematics, so it will turn out that constructive
mathematics is inadequate for the needs of core theoretical
physics.
Volker Halbach has asked me privately if ACA_0 would be
sufficient for theoretical physics. I doubt it. By known results, it
can’t prove such things as Ramsey’s Theorem (Simpson 1998, pp.
210-215), and it is perfectly conceivable that combinatorial stuff
might well be required in theoretical physics. It can’t prove the
Sigma^1_0 axiom of choice (you need ATR_0: Simpson 1998, pp.,
205-206) and that might be needed in physics.
Moreover, I think that something like “reverse physics” could be
developed. I’ll discuss this a bit more below (Section (L)).
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(F) The Kazhdan problem: the difference between interesting and
uninteresting mathematical structures.
Mark Steiner mentions that core mathematicians wouldn’t treat
“the structure” of chess as a mathematical structure, ans cites
Kazhdan. But that’s really a separate problem, about what it is that
makes some structures more *interesting* than others. I don’t think
they would say “Chess has no structure at all”. I think they would
say “OK. It’s a structure, but a very boring one”.
I don’t know why some mathematical structures are more
interesting to study in their own right than others. But it’s a bit like
asking why physicists don’t study, say, pebbles on the beech.
Pebbles are physical systems. Why not study them? Well, the
fact that physicists don’t take much interest in studying pebbles
doesn’t imply that they don’t, in general, study the behaviour of
physical systems.
(By the way, I remember Feynman once describing in a TV
interview how, one drunk night, he found himself making spaghetti
and he noticed an interesting pattern in the way it snapped when
bent. He spent a couple of hours trying to figure out physically why
this happens, and then gave up).
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(G) The Wigner problem: The “unreasonable applicability” problem
Mark Steiner discusses this problem in excellent detail in his book
and in his posting. An example I would add to his list is Dirac’s
analogy between Poisson brackets in classical mechanics, which
are then taken over as commutation relations to quantum
mechanics. In classical physics, {q, p} = 0. In quanatum physics,
[Q, P] = ih.
[By the way, the quantum commutation relations [Q, P] = ih have
no countable representations. The spectrum of the self-adjoint
operators Q and P are forced (by the commutation relations) to be
the whole real line. That demolishes the often-canvassed idea that
quantum mechanics somehow suggest that space should be
"discrete"].
Maybe the universe is mathematical in a way that is closely
connected to the way that the human mind itself is mathematical,
and so purely abstract structures that strike us as interesting to
study (like SU(2)) are precisely those that appear in nature. But I
really have no idea. It’s a big problem.
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(H) The Kronecker problem: The finiteness of the human mind.
The mathematician Leopold Kronecker was a famous early finitist,
saying something like “God created the integers and the rest is
Man’s work”. The allegedly finite nature of the mind is a massive
topic, but Matt Insall asked about why we usually assume that the
mind is finite (for the human brain takes up, say, 100 cubic cm of
physical space, and that contains uncountably many spacetime
points).
First, most agree these days that the mind is somehow “produced”
by the brain. (But, of course, no one knows how). Second, the
alleged finiteness of the mind is an empirical matter (and
presumably contingent matter: we can imagine a possible world
containing beings whose minds weren’t finite, and who could
perform mental supertasks). But this is an assumption built into
almost all discussions of the relation between the mind and
computational models of the mind.
Turing himself (I think) said that, in developing the concept of a
Turing machine, he had wanted to model the way an *idealized
mind* would think when computing the solution to a problem, using
pencil and paper. Flashes of “insight” or “intuition” are not allowed.
Chomsky (1957: Syntactic Structures) asked the question: how
can the finite mind grasp an infinite number of meaningful
sentences? He often mentions that this question had already
occurred to von Humboldt, a 100 years before. One of his great
insights was to use the notions of recursion and computability
developed 20 years before by Godel, Post, Church and Turing. The
set of meaningful sentences in humanly learnable language must
be a decidable set. It is the standard view in theoretical lingusitics
that a language containing a non-recursive set of meaningful
sentences recursive would simply be unlearnable.
But it’s an empirical matter all the same.
Roger Penrose does believe that there are flashes of “insight” or
“mathematical intuition” which cannot be modelled computationally
or algorithmically. He cites the apparent human ability to recognize
that the Godel sentence for Peano Arithmetic is true, even though
G isn’t a theorem of PA. Of course, many, many people have
written about this (including Martin Davis and even me), disputing
the validity of this argument.
I don’t think that the fact that the human cranium encloses
uncountably many spacetime points helps with this problem. There
is a basic neuronal level of brain activity – neurotransmitters moving
along synapses -- and I am very sceptical that the infinity in
question could be “harnessed” by brain cells. Penrose thinks some
funny quantum process, combing quantum gravity with a realistic
interpretation of wavefunction collapse, is at work in generating
human consciousness. Mmmm.
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(I) The Laplace-Poincare problem: predictability and computational
tractability.
Laplace proposed that a creature which knew all the present
conditions of the universe and its laws, could predict all future
events and states. Poincare studied the predictability of n-body
problems in classical phase space. There are many topics here,
many of clear fom relevance. Joe Shipman has sent me his paper,
[12] Aspects of Computability in Physics
This is one area where there has been some important research. In
particular, there is the work of Pour-El and Richards
[13] Pour-El and Richards 1987: Computability and Analysis in
Physics
Results in this area are concerned with how (classical) dynamical
systems evolve over time. In particular, Pour-El and Richards
showed that for the conventional wave equation, a computable real-
valued function describing the initial data could evolve into a non-
computable function. I don’t know much more than that really.
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(J) Quantum computation
Matt Insall mentioned quantum computation in his posting. I don’t
know too much about this topic either. David Deutsch – one of the
pioneers of the idea -- has written a book (which I don’t possess
unfortunately), including a discussion of this stuff, called
[14] David Deutsch 1998: The Fabric of Reality
Deutsch thinks that the possibility of quantum computation points
towards to so-called many universes interpretation of quantum
mechanics.
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(K) The Zeno Problem: supertasks in spacetime and infinity
machines
I think there’s an even more interesting development than quantum
computation. That’s the possibility of supertasks in physical
spacetime, an idea which goes right back to Zeno’s paradoxes. A
supertask is an infinite sequence of distinct operations,
completed in a finite time. A Turing machine which doesn’t halt
performs a supertask, but unfortunately takes infinitely long to
“finish” its business (of course, there is no
final state). “Counting” through all the numbers to see if and
equation of the form f(n) = 0 has a solution is a supertask.
Recently a young philosopher at Cambridge, Mark Hogarth, has
considered unusual space-times (now called Malament-Hogarth
spacetimes) in which an infinitely long geodesic (i.e., which has an
infinite proper time) can be viewed in its entirety by an observer
over some finite proper time interval. Put the Turing machine M on
this geodesic and wait until its “finished” (strictly speaking, the
information you receive is either (i) M halted at some finite stage n
or (ii) M actually didn’t halt at all. It is this latter information which
is not available in the usual case). You have what Earman
and Norton call an “infinity machine”. See:
[15] Earman and Norton 1996, “Infinite Pains: The Trouble with
Supertasks”, in Morton (ed.) 1996. Benacerraf and his Critics.
This paper is an excellent introduction to supertasks, debunking
some of the alleged paradoxes associated with supertasks, and
discussing the possible implications of building “infinity machines”.
In particular, using simple infinity machines, the decision problem
for Pi^0_1 and Sigma^0_1 sentences is soluble, by getting the
machine to run through all the numbers, looking for verifiers or
falsifiers. For more complex formulas, you need an infinity of infinity
machines, all chained together. Apparently this is discussed in,
[16] Mark Hogarth 1994: “Relativistic non-Turing machines and the
failure of Church’s Thesis” (don't know where this appeared)
I’m not sure, but can’t the consistency of formal system be
encoded as a Pi^0_1 sentence? If so, then presumably such a
simple infinity machine could prove the consistency of any formal
system, such as ZFC. The infinity machine simply determines the
truth value of the Pi^0_1 sentence coding ZFC’s consistency. If the
infinity machine doesn’t in fact halt (after its infinity of operations),
then ZFC is, in fact, consistent.
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(L) Godelian phenomena in physics: Reverse physics?
Physical theories are really no different from mathematical
theories, except that mathematicians have made much greater
efforts to formalize and study mathematical theories proof-
theoretically and model-theoretically. In particular, physical
theories can suffer from Godel-type incompleteness (if the universe
is rich enough to contain a spacetime model of natural numbers,
which is probably true). This means that GR, for example, will be
deductively incomplete w.r.t. the sentences about spacetime.
Adding say CH or a large cardinal axiom to GR may then generate
some new consequences.
One might conceive of a kind of “reverse physics” with some basic
geometrical spacetime theory Geom as its base theory. Then one
might try to discover “reverse physics” results of the form:
(i) Geom |- GH <-> A
Where A is some (preferably experimentally measureable)
statement about spacetime. If we could determine, by experiment,
that A is true, then that would be an excellent reason for accepting
CH, even though a reason derived from physical experiment.
This would be completely analogous to a standard reverse maths
result, say:
(ii) RCA_0 |- ACA_0 <-> BW
Where BW is the Bolzano-Weierstrass Theorem.
Jeff Ketland
Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel: 0115 951 5843
Fax: 0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>
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